Finite Group Databasebeta

← #115488

Subgroups — #115488

Order 512.

1158 conjugacy classes of subgroups.

The subgroup lattice has 631 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#477C211yesmaximal, characteristic
2C2×#1987C211yesmaximal, characteristic
3C2×#1956C211yesmaximal, characteristic
4C22×Q8⋊C8C211yesmaximal, characteristic
5#13926C211yesmaximal, characteristic
6C2×#1296C211yesmaximal, characteristic
7C2×#1279C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10C2×#477—22no—
11C2×#477—22no—
12C2×#477—22no—
13C2×#477—22no—
14C2×#477—22no—
15C2×#477C414yescomplemented
16C22×#107—21no—
17C22×C4⋊C8C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C22×#103—21no—
20C22×#103—22no—
21C22×C4×C8C2211yescharacteristic
22C2×#391—21no—
23C2×#384C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25#1987—22no—
26#1987—22no—
27#1956—22no—
28#1956—22no—
29C2×#340—22no—
30C2×Q8⋊C8—22no—
31C2×Q8⋊C8—22no—
32C2×Q8⋊C8—22no—
33#1441—22no—
34C4⋊#95—22no—
35C4⋊#95—21no—
36#1296—22no—
37#1279—22no—
38C23×Q8—21no—
39C23×Q8D411yescharacteristic
40C23×C8—21no—
41C23×C8—21no—
42C23×C8—21no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8C2×C412yes—
53C2×C4⋊Q8D412yes—
54C2×C4⋊Q8D412yes—
55C2×C4×Q8—21no—
56C2×C4×Q8—21no—
57C2×C4×Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8—21no—
61C2×C4×Q8—21no—
62C2×C4×Q8—21no—
63C2×C4×Q8—21no—
64C2×C4×Q8—21no—
65C2×C4×Q8C2×C412yes—
66C2×C4×Q8D412yes—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4D412yes—
74C22×C4⋊C4D412yes—
75C22×C42—21no—
76C22×C42—21no—
77C22×C42C2311yescharacteristic
78#477—28no—
79#477—28no—
80#477—28no—
81#477—28no—
82C2×#107—24no—
83C2×#107—24no—
84C2×#107—24no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8C2×C412yes—
94C2×C4⋊C8D412yes—
95C2×C4⋊C8D412yes—
96C2×C4⋊C8D412yes—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103—28no—
101C2×#103—48no—
102C2×C4×C8—21no—
103C2×C4×C8—21no—
104C2×C4×C8—21no—
105C2×C4×C8—21no—
106C2×C4×C8—21no—
107C2×C4×C8D412yes—
108#391—42no—
109#391—42no—
110#384—21no—
111#384—21no—
112#384—21no—
113#384—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C2×#95—24no—
119C2×#95—21no—
120C2×#95—24no—
121#340—28no—
122Q8⋊C8—28no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8—22no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—22no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—22no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—44no—
138C22×Q8—44no—
139C22×Q8—41no—
140C22×Q8—41no—
141C22×Q8—41no—
142C22×Q8C22⋊C412yes—
143C22×Q8SD1612yes—
144C22×Q8Q1612yes—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—41no—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C22×C8—24no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—42no—
160C22×C8—41no—
161C22×C8—41no—
162C22×C8—41no—
163C22×C8—41no—
164C22×C8—41no—
165C22×C8—41no—
166C22×C8—41no—
167C22×C8—41no—
168C22×C8—42no—
169C22×C8—42no—
170C4⋊Q8—22no—
171C4⋊Q8—22no—
172C4⋊Q8—22no—
173C4⋊Q8—22no—
174C4⋊Q8—22no—
175C4⋊Q8—22no—
176C4⋊Q8—22no—
177C4⋊Q8—22no—
178C4⋊Q8—22no—
179C4⋊Q8—41no—
180C4⋊Q8—41no—
181C4⋊Q8—41no—
182C4⋊Q8—41no—
183C4⋊Q8—41no—
184C4⋊Q8—41no—
185C4⋊Q8C22⋊C414yes—
186C4×Q8—22no—
187C4×Q8—22no—
188C4×Q8—22no—
189C4×Q8—22no—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×Q8—22no—
193C4×Q8—22no—
194C4×Q8—41no—
195C4×Q8—41no—
196C4×Q8—41no—
197C4×Q8—41no—
198C4×Q8—41no—
199C4×Q8—41no—
200C4×Q8—41no—
201C4×Q8—41no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4C22⋊C412yes—
244C2×C4⋊C4Q1614yes—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—41no—
248C2×C42—41no—
249C2×C42—41no—
250C2×C42—41no—
251C2×C42—41no—
252C2×C42—41no—
253C2×C42—41no—
254C2×C42—41no—
255C2×C42—41no—
256C2×C42—41no—
257C2×C42C22×C411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42#4111yescharacteristic
266C2×C42#4111yescharacteristic
267C2×C42#4111yescharacteristic
268C2×C42#4111yescharacteristic
269#107—216no—
270C4⋊C8—22no—
271C4⋊C8—22no—
272C4⋊C8—22no—
273C4⋊C8—22no—
274C4⋊C8—22no—
275C4⋊C8—41no—
276C4⋊C8—41no—
277C4⋊C8C22⋊C414yes—
278#103—216no—
279#103—416no—
280C4×C8—22no—
281C4×C8—22no—
282C4×C8—41no—
283C4×C8—41no—
284#95—48no—
285#95—41no—
286#95—41no—
287#95—84no—
288C24#12011yescharacteristic
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—24no—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—44no—
306C2×Q8—48no—
307C2×Q8—44no—
308C2×Q8—48no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—41no—
312C2×Q8—41no—
313C2×Q8—42no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×Q8—41no—
317C2×Q8—41no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×Q8—41no—
321C2×Q8—41no—
322C2×Q8—41no—
323C2×Q8—41no—
324C2×Q8—84no—
325C2×Q8—82no—
326C2×Q8—82no—
327C2×Q8—84no—
328C2×Q8#10314yescomplemented
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—82no—
359C22×C4—82no—
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C4×D411yescharacteristic
362C22×C4C4×D411yescharacteristic
363C22×C4#12011yescharacteristic
364C22×C4#12011yescharacteristic
365C22×C4#12011yescharacteristic
366C22×C4#12011yescharacteristic
367C22×C4#12111yescharacteristic
368C22×C4#12111yescharacteristic
369C22×C4#12111yescharacteristic
370C22×C4#12111yescharacteristic
371C22×C4#12111yescharacteristic
372C22×C4#12111yescharacteristic
373C22×C4#12111yescharacteristic
374C22×C4#12111yescharacteristic
375C22×C4#12311yescharacteristic
376C22×C4#12711yescharacteristic
377C22×C4#12711yescharacteristic
378C22×C4C2×SD1611yescharacteristic
379C22×C4C2×Q1611yescharacteristic
380C22×C4C2×Q1612yes—
381C22×C4#13511yescharacteristic
382C22×C4#13511yescharacteristic
383C22×C4#13512yes—
384C22×C4AGL(1,ℤ/8)11yescharacteristic
385C22×C4#13711yescharacteristic
386C22×C4#13711yescharacteristic
387C22×C4#13711yescharacteristic
388C22×C4#13712yes—
389C22×C4#13712yes—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—44no—
393C2×C8—42no—
394C2×C8—44no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—44no—
404C2×C8—82no—
405C2×C8—81no—
406C2×C8—81no—
407C4⋊C4—24no—
408C4⋊C4—22no—
409C4⋊C4—24no—
410C4⋊C4—22no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—44no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—82no—
429C4⋊C4—81no—
430C4⋊C4—81no—
431C4⋊C4—82no—
432C4⋊C4—81no—
433C4⋊C4—81no—
434C4⋊C4—81no—
435C4⋊C4—81no—
436C42—42no—
437C42—42no—
438C42—42no—
439C42—42no—
440C42—81no—
441C42—81no—
442C42—81no—
443C42—81no—
444C42C2×C22⋊C411yescharacteristic
445C42C2×C22⋊C411yescharacteristic
446C42C4×D411yescharacteristic
447C42C4×D411yescharacteristic
448C42#12011yescharacteristic
449C42#12011yescharacteristic
450C42#12011yescharacteristic
451C42#12111yescharacteristic
452C42#12111yescharacteristic
453C42#12111yescharacteristic
454C42#12111yescharacteristic
455C42#12311yescharacteristic
456C42#12311yescharacteristic
457C42#12311yescharacteristic
458C42#12411yescharacteristic
459C42#12411yescharacteristic
460C23#38611yescharacteristic
461C23#39211yescharacteristic
462C23#39211yescharacteristic
463C23#45011yescharacteristic
464C23#45111yescharacteristic
465C23#45112yes—
466C23#45211yescharacteristic
467C23#45211yescharacteristic
468C23#45212yes—
469C23#45312yes—
470C23#45412yes—
471Q8—24no—
472Q8—24no—
473Q8—48no—
474Q8—42no—
475Q8—42no—
476Q8—42no—
477Q8—42no—
478Q8—42no—
479Q8—42no—
480Q8—84no—
481Q8—84no—
482Q8—84no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—84no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—81no—
509C2×C4—82no—
510C2×C4—81no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—84no—
515C2×C4—84no—
516C2×C4—84no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4#37911yescharacteristic
521C2×C4#38611yescharacteristic
522C2×C4#38611yescharacteristic
523C2×C4#38611yescharacteristic
524C2×C4#39011yescharacteristic
525C2×C4#39011yescharacteristic
526C2×C4#39211yescharacteristic
527C2×C4#39211yescharacteristic
528C2×C4#39411yescharacteristic
529C2×C4#39411yescharacteristic
530C2×C4#39411yescharacteristic
531C2×C4#39411yescharacteristic
532C2×C4C2×#10311yescharacteristic
533C2×C4#41611yescharacteristic
534C2×C4#41711yescharacteristic
535C2×C4#41911yescharacteristic
536C2×C4C4×Q1612yes—
537C2×C4#44112yes—
538C2×C4#44811yescharacteristic
539C2×C4#44811yescharacteristic
540C2×C4#44911yescharacteristic
541C2×C4#44911yescharacteristic
542C2×C4#45112yes—
543C2×C4#45111yescharacteristic
544C2×C4#45111yescharacteristic
545C2×C4#45212yes—
546C2×C4#45211yescharacteristic
547C2×C4#45211yescharacteristic
548C2×C4#46212yes—
549C2×C4#46212yes—
550C2×C4#46412yes—
551C2×C4#46412yes—
552C2×C4#46511yescharacteristic
553C2×C4#46511yescharacteristic
554C2×C4#46712yes—
555C2×C4#46711yescharacteristic
556C2×C4#46711yescharacteristic
557C2×C4#46811yescharacteristic
558C2×C4#46811yescharacteristic
559C2×C4#47012yes—
560C2×C4#47011yescharacteristic
561C2×C4#47011yescharacteristic
562C2×C4#48311yescharacteristic
563C2×C4#48411yescharacteristic
564C2×C4#48511yescharacteristic
565C2×C4#48511yescharacteristic
566C2×C4#49411yescharacteristic
567C2×C4#49511yescharacteristic
568C2×C4#49711yescharacteristic
569C2×C4#49711yescharacteristic
570C8—44no—
571C8—44no—
572C8—82no—
573C22#143811yescharacteristic
574C22#163012yes—
575C22#163112yes—
576C22#163212yes—
577C22#163312yes—
578C22#163612yes—
579C22#163612yes—
580C22#163712yes—
581C22#163812yes—
582C22#187211yescharacteristic
583C22#187311yescharacteristic
584C22#187812yes—
585C22#188312yes—
586C22#200011yescharacteristic
587C22#200011yescharacteristic
588C22#200112yes—
589C22#200111yescharacteristic
590C22#200111yescharacteristic
591C22#200211yescharacteristic
592C22#200311yescharacteristic
593C22#200612yes—
594C22#201511yescharacteristic
595C22#201611yescharacteristic
596C4—21no—
597C4—21no—
598C4—21no—
599C4—21no—
600C4—44no—
601C4—82no—
602C4—84no—
603C4—84no—
604C4—162no—
605C4#178711yescharacteristic
606C4#178811yescharacteristic
607C4#187812yes—
608C4#189211yescharacteristic
609C4#189411yescharacteristic
610C4#196911yescharacteristic
611C4#197111yescharacteristic
612C4#201511yescharacteristic
613C4#201611yescharacteristic
614C4#203511yescharacteristic
615C4#203511yescharacteristic
616C4#203611yescharacteristic
617C4#203611yescharacteristic
618C4#205612yes—
619C2#897412yes—
620C2#897512yes—
621C2#1005511yescharacteristic
622C2#1005711yescharacteristic
623C2#1005911yescharacteristic
624C2#1006111yescharacteristic
625C2#1023612yes—
626C2#1023712yes—
627C2#1097011yescharacteristic
628C2#1101611yescharacteristic
629C2#1102411yescharacteristic
subgroupG/H
center, socle, prime radicalC24#120
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
15. C2×#477C4semidirect
328. C2×Q8#103semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#4772
C2×#19872
C2×#19562
C22×Q8⋊C82
#139262
C2×#12962
C2×#12792